Field of the Disclosure
The present disclosure relates generally to constructing rate-compatible polar codes, and more particularly, to constructing hybrid automatic repeat request (HARQ) rate-compatible polar codes for wireless channels.
Description of the Related Art
Polar codes are the first and currently only family of codes with explicit construction (i.e., no ensemble to pick from) and low-complexity encoding and decoding algorithms that achieve the capacity over a certain class of channels. A polar transformation is defined as a multiplication of an input vector by a polarization matrix
      F    2          ⊗      n        =                    [                                            1                                      0                                                          1                                      1                                      ]                    ⊗        n              .  
Polar code construction, or channel polarization, is based on the observation that as a length N=2n of a polar transformation increases, the observed bit-channels at the input are polarized so that they become either noiseless (perfect) channels or completely noisy channels. A polar code is constructed by transmitting information bits over the noiseless channels, also referred to as good bit-channels, while restricting (or freezing) the input to the noisy channels, also referred to as bad bit-channels, to zeros.
Constructing polar codes (i.e., finding good bit-channels) is, in general, a hard problem. There are some heuristic and approximate algorithms that attempt to solve the problem. However, they only concern one given channel and do not disclose a method of constructing a polar code that is universally good for an arbitrary class of channels.
Another complicating factor is that the construction of polar codes, in general, depends on the characteristics of the underlying channel. As a result, if a polar code is optimized for transmission across a certain channel, it may not be good for transmission over another channel. This is a challenge in constructing polar codes suitable for practical applications, because in a communications system, an underlying channel varies. Thus, there is a need for an apparatus and method of constructing a polar code that is robust to channel variations.
Wireless systems also suffer from uncertainties in channel estimation and delayed feedback. To achieve better overall system throughput, a hybrid automatic repeat request (HARQ) protocol is often deployed. In a HARQ system, data may be encoded with an error detection code, such as cyclic redundancy check code, for error detection control. The data may also be encoded with an error correction code, such as a polar code, to improve the error rate performance.
Upon receipt of a transmission, the error detection control mechanism is checked. If the error detection control mechanism indicates that a transmission was received successfully, then an acknowledgement message (ACK) is generated, and no further transmissions are required. If the error detection mechanism indicates that a transmission was not received successfully, then a non-acknowledgement message (NACK) is generated, and a retransmission is requested from the transmitter node. Retransmissions can be the exact same data bits as the previous transmission (commonly referred to as Chase combining), or can include new redundancy bits, commonly referred to as incremental redundancy (IR), which when combined with data of the previous transmission form another codeword having a lower rate, where a rate of a code (k/N) is a ratio of information bits (k) in a code to the total number of bits (N) in the code. In hybrid Chase/IR transmissions, retransmissions can include some of the previously transmitted bits as well as some new redundancy bits.
There is a need for a rate-compatible family of codes that are suitable for HARQ transmissions.